Course Archives Theoretical Statistics and Mathematics Unit | |||
Course: Measure Theory Level: Postgraduate Time: Currently not offered |
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Syllabus Past Exams Syllabus: i) The concept of s-algebra, Borel subsets of R, Construction of Lebesgue and Lebesgue-Stieltjes measures on the real line following outer measure. ii) Abstract measure theory: definition and examples of measure space, measur- able functions, Lebesgue integration, convergence theorems (Fatous Lemma, Monotone convergence and dominated convergence theorem). iii) Caratheodory extension theorem, completion of measure spaces. iv) Product measures and Fubinis theorem. v) Lp-spaces, Riesz-Fischer Theorem, approximation by step functions and con- tinuous functions. vi) Absolute continuity, Hahn-Jordan decomposition, Radon-Nikodym theorem, Lebesgue decomposition theorem. Functions of bounded variation. vii) Complex measures. If time permits: Vitali covering lemma, differentiation and fundamental theorem of calculus. Suggested Texts : (a) H. L. Royden and Patrick Fitzpatrick: Real Analysis, Pearson, 4th edition. (b) Robert B. Ash and Catherine A. Doleans-Dade, Probability and measure theory, GTM(211), Academic Press, 2nd edition. (c) Elias M. Stein, Rami Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces , Princeton Lectures in Analysis. (d) Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, A Wiley Series. (e) G. de Barra, Measure Theory and Integration. Top of the page Past Exams | |||
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